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Classical change point analysis aims at (1) detecting abrupt changes in themean of a possibly non-stationary time series and at (2) identifying regionswhere the mean exhibits a piecewise constant behavior. In many applicationshowever, it is more reasonable to assume that the mean changes gradually in asmooth way. Those gradual changes may either be non-relevant (i.e., small), orrelevant for a specific problem at hand, and the present paper presentsstatistical methodology to detect the latter. More precisely, we consider thecommon nonparametric regression model $X_{i} = \mu (i/n) + \varepsilon_{i}$with possibly non-stationary errors and propose a test for the null hypothesisthat the maximum absolute deviation of the regression function $\mu$ from afunctional $g (\mu )$ (such as the value $\mu (0)$ or the integral$\int_{0}^{1} \mu (t) dt$) is smaller than a given threshold on a giveninterval $[x_{0},x_{1}] \subseteq [0,1]$. A test for this type of hypotheses isdeveloped using an appropriate estimator, say $\hat d_{\infty, n}$, for themaximum deviation $ d_{\infty}= \sup_{t \in [x_{0},x_{1}]} |\mu (t) - g( \mu)|$. We derive the limiting distribution of an appropriately standardizedversion of $\hat d_{\infty,n}$, where the standardization depends on theLebesgue measure of the set of extremal points of the function$\mu(\cdot)-g(\mu)$. A refined procedure based on an estimate of this set isdeveloped and its consistency is proved. The results are illustrated by meansof a simulation study and a data example.

Are deviations in a gradually varying mean relevant? A testing approach based on sup-norm estimators